A long time ago, John Conway posted a very thoughtful responseto a posting of mine about mathematical habits of mind. Belowis a belated response from my project. I would ver much like tocontinue this discussion. Thoughtful criticism from intelligentpeople is the best way to improve the work we're doing.

`` However, there is something about Michelle's paper thatworries me TREMENDOUSLY. I'm sure it's mostly accidental, butit doesn't seem to mention what I regard as the most importantthing of all about the mathematical experience, from both thepractical and theoretical standpoints.

This is the habit of PRECISE thinking about PRECISELY wordedproblems. This is the most important thing to teach, and shouldprecede ANY kind of thinking about FUZZILY worded problems, in myview (and I'm speaking here as a teacher, rather than as aprofessional mathematician).''

He is right (of course); his worry is due to an accidental slip (or, moreprecisely, to a lack of a precise description of what we mean) in the paper.

Here's an annotated version of the paragraph in question:

``There is another way to think about it, and it involves turning the priorities around. Much more important than specific mathematical results are the habits of mind used by the people who create those results, and we envision a curriculumthat elevates the methods by which mathematics is created, thetechniques used by researchers, to a status equal to that enjoyedby the results of that research.

[This includes (1) precise thinking about precise problems and(2) precise thinking abut not-so precise problems. It also includes using(3) heuristics and intuition to come up with plausible conjectures. Examples from some curriculum materials we are developing: (1) Students are given the explicit task of cutting a rectangle up to formanother rectangle on a different base. (2) Students are asked to definewhat they mean by ``best'' if the want to find the best spot for anairport that will serve three cities. (3) Students are asked to imagine aline segment, starting at one base of a trapezoid, moving up parallel tothat base (connecting points on the non-parallel sides), stopping at thetop base, and are then asked to use this thought experiment toconjecture a general formula for the median of a trapezoid.]

The goal is not to train largenumbers of high school students to be university mathematicians,but rather to allow high school students to become comfortablewith ill-posed and fuzzy problems, to see the benefit ofsystematizing and abstraction, and to look for and develop newways of describing situations. ''

[ We should have emphasized precision when we wrote ``to see thebenefit of systematizing and abstraction, and to look for and developnew ways of describing situations;'' it was certainly on our minds.]

Right after that,we say:

``While it *is* necessary toinfuse courses and curricula with modern content, what's evenmore important is to give students the tools they'll need touse, understand, and even make mathematics that doesn't yetexist.''

A curriculum organized around habits of mind tries to close the gap between what the users and makers of mathematics *do*and what they *say*. Such a curriculum lets students inon the process of creating, inventing, conjecturing, andexperimenting; it lets them experience what goes on behind thestudy door *before* new results are polished and presented. It is a curriculum that encourages false starts, calculations, experiments, and special cases. Students develop the habit of reducing things to lemmas for which they have no proofs, suspending work on these lemmas andon other details until they see if assuming the lemmas willhelp. It helps students look for logical and heuristicconnections between new ideas and old ones. A habits of mindcurriculum is devoted to giving students a genuine researchexperience.

[There are two main thrusts of the paper: (1) to call for a mathematicsexperience based on the *interplay* between deduction and experimentthat is so crucial to doing mathematics (that's the integration ofthe logical (precise) and heuristic (less-precise) ways of thinking) and(2) to concentrate on the way mathematics is *developed* as opposedto the way it is presented. For example, students are introducedto proof as a method for communication *and* as atechnique for discovery. The communication of an argument is arather precise activity, and we show students a few ways to do this(two-column, paragraph, as well as some presentation techniques fromRussia, China, and Israel). The *search* for an argument is a morefuzzy activity, which makes it harder to teach, but we spend a greatdeal of time developing some general principles for navigating throughthe morass of detail that you face when trying to establish aconjecture.]

So, the point is: we certainly agree that precision and precise thinking are central to mathematics, and, if that didn't come across in the paper,it's due more to shoddy writing than to shoddy intent. The paper willbe appearing soon in JMB. Maybe some letters to the editor published insubsequent issues can clear this up.